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Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group

Part of: Holomorphic fiber spaces Low-dimensional topology (Co)homology theory

Published online by Cambridge University Press:  19 April 2024

Luca F. Di Cerbo  and
Luigi Lombardi
Luca F. Di Cerbo
Affiliation:
Mathematics Department, University of Florida, Gainesville, FL, USA ([email protected])
Luigi Lombardi
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano Statale, via Cesare Saldini 50, Milan 20133, Italy ([email protected])
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Abstract

We prove the Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group.

Keywords

Albanese mapL2-Betti numbersSinger conjecturenormalized Betti numbersfundamental group

MSC classification

Primary: 14F45: Topological properties
Secondary: 32L20: Vanishing theorems 57M07: Topological methods in group theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 7
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Abert, M., Bergeron, N., Biringer, I. and Gelander, T.. Convergence of normalized Betti numbers in nonpositive curvature. Duke Math. J. 172 (2023), 633–700.CrossRefGoogle Scholar
Avramidi, G., Okun, B. and Schreve, K.. Mod $p$ and torsion homology growth in nonpositive curvature. Invent. Math. 226 (2021), 711–723.CrossRefGoogle Scholar
Atiyah, M. F.. Elliptic operators, discrete groups and von Neumann algebras. In Colloque ‘Analyse et Topologie’ en l'Honneur de Henri Cartan (Orsay, 1974). Astérisque, vol. 32–33, pp. 43–72 (Paris: Soc. Math. France, 1976).Google Scholar
Beauville, A.. Complex Algebraic Surfaces, 2nd edn. London Mathematical Society Student Texts, vol. 34 (Cambridge: Cambridge University Press, 1996).Google Scholar
Di Cerbo, G. and Di Cerbo, L. F.. On Seshadri constants of varieties with large fundamental group. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), 335–344.Google Scholar
Di Cerbo, L. F. and Lombardi, L.. $L^2$-Betti numbers and convergence of normalized Hodge numbers via the weak generic Nakano vanishing theorem. Ann. Inst. Fourier (2023), 27. (Online first).Google Scholar
Di Cerbo, L. F. and Stern, M.. Price inequalities and Betti number growth on manifolds without conjugate points. Commun. Anal. Geom. 30 (2022), 297–334.CrossRefGoogle Scholar
Dodziuk, J.. $L^{2}$ harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Am. Math. Soc. 77 (1979), 395–400.Google Scholar
Donnelly, H. and Xavier, F.. On the differential form spectrum of negatively curved Riemannian manifolds. Am. J. Math. 106 (1984), 169–185.CrossRefGoogle Scholar
Gromov, M.. Kähler hyperbolicity and $L_2$-Hodge theory. J. Differ. Geom. 33 (1991), 263–292.CrossRefGoogle Scholar
Jost, J. and Zuo, K.. Vanishing theorems for $L^2$-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry. Commun. Anal. Geom. 8 (2000), 1–30.CrossRefGoogle Scholar
Lück, W.. Approximating $L^2$-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4 (1994), 455–481.CrossRefGoogle Scholar
Lück, W.. $L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 44 (Berlin: Springer-Verlag, 2002).Google Scholar
Liu, Y., Maxim, L. and Wang, B.. Aspherical manifolds, Mellin transformation and a question of Bobadilla-Kollár. J. Reine Angew. Math. 781 (2021), 1–18.CrossRefGoogle Scholar
Liu, Y., Maxim, L. and Wang, B.. Topology of subvarieties of complex semi-abelian varieties. Int. Math. Res. Not. 14 (2021), 11169–11208.CrossRefGoogle Scholar
Schoen, R. and Yau, S.-T.. Lectures on differential geometry. In Conference Proceedings and Lecture Notes in Geometry and Topology, vol. I (Cambridge, MA: International Press, 1994).Google Scholar